Algebraic CMI decay and improved boundary dependence for unrestricted-support power-law interactions

Establish algebraic conditional mutual information decay for quantum Gibbs states with unrestricted-support power-law interactions throughout the regime D<ζ≤2D, improve the distance-decay exponents obtained for power-law interactions, and establish stronger boundary dependence in the resulting bounds.

Background

The paper proves conditional mutual information bounds for quantum Gibbs states with unrestricted-support power-law interactions under an aggregate pair-decay assumption with exponent ζ>D. In the regime D<ζ≤2D, the established bounds are inverse-logarithmic, while algebraic decay is obtained only when ζ>2D. The authors identify the threshold 2D as sufficient for their truncation argument rather than necessarily optimal.

The open directions concern extending algebraic decay to the full summable regime D<ζ≤2D, improving the distance exponents beyond those supplied by the current modular-tail and spatial-truncation analysis, and replacing the present polynomial or volume-related region dependence by stronger dependence on the boundary of the region. These are unresolved extensions of the paper's power-law local-Markovianity results.